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Groupes de Picard et problèmes de Skolem. I. (Picard groups and Skolem problems. I). (French) Zbl 0704.14014

A problem of Skolem asks about the existence of a solution of a system of diophantine equations over a ring R of algebraic integers whose coordinates belong to a finite extension of R. In the scheme-theoretical language this is a question about the existence of a multi-section of a morphism of finite type \(f:X\to Spec(R)\), where R is a Dedekind ring.
The main result of this paper asserts that the answer is positive if the following conditions are satisfied:
(i) R is an excellent ring whose residue fields are algebraic over a finite field;
(ii) for any finite extension \(K'\) of the field of fractions K of R the normalization \(R'\) of R in \(K'\) has torsion Picard group;
(iii) there exists \(K'\) as above such that one of the irreducible components of the base change \(X\otimes_ RR'\) is mapped surjectively to \(Spec(R').\)
This result is a generalization of a theorem of R. S. Rumely [J. Reine Angew. Math. 368, 127-133 (1986; Zbl 0581.14014)], who assumed that R is a ring of integers in an algebraic number field and f is surjective. The proof is geometric and does not use the theory of capacity of Rumely.
[See also the following review.]
Reviewer: I.V.Dolgachev

MSC:

14G25 Global ground fields in algebraic geometry
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
11R04 Algebraic numbers; rings of algebraic integers
11D41 Higher degree equations; Fermat’s equation
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